scholarly journals Generating Functions of Binary Products of Tribonacci and Tribonacci Lucas Polynomials and Special Numbers

2022 ◽  
pp. 237-262
Author(s):  
Hind Merzouk ◽  
Ali Boussayoud ◽  
Kasi Viswanadh V. Kanuri
Keyword(s):  
Generating Functions ◽  
Lucas Polynomials

In this paper, we introduce a new operator defined in this paper, we give some new generating functions of binary products of Tribonacci and Tribonacci Lucas polynomials and special numbers.

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

scholarly journals Symmetric and generating functions of generalized (p,q)-numbers

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
Nabiha Saba ◽  
◽  
Ali Boussayoud ◽  
Abdelhamid Abderrezzak ◽  
◽  
...  
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Lucas Polynomials ◽  
Binet’S Formula

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

scholarly journals On the (p,q)–Chebyshev Polynomials and Related Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 136 ◽  
Author(s):  
Can Kızılateş ◽  
Naim Tuğlu ◽  
Bayram Çekim
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Lucas Polynomials

In this paper, we introduce ( p , q ) –Chebyshev polynomials of the first and second kind that reduces the ( p , q ) –Fibonacci and the ( p , q ) –Lucas polynomials. These polynomials have explicit forms and generating functions are given. Then, derivative properties between these first and second kind polynomials, determinant representations, multilateral and multilinear generating functions are derived.

scholarly journals Parity Theorems for Statistics on Domino Arrangements

10.37236/1977 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark A. Shattuck ◽  
Carl G. Wagner
Keyword(s):  
Generating Functions ◽  
Lucas Polynomials ◽  
Special Values

We study special values of Carlitz's $q$-Fibonacci and $q$-Lucas polynomials $F_n(q,t)$ and $L_n(q,t)$. Brief algebraic and detailed combinatorial treatments are presented, the latter based on the fact that these polynomials are bivariate generating functions for a pair of statistics defined, respectively, on linear and circular domino arrangements.

Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials and their properties

2020 ◽  
pp. 2150076
Author(s):  
Gamaliel Cerda-Morales
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Third Order ◽  
Lucas Polynomials ◽  
Binet Formula

In this paper, the Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are obtained by using the recurrence relation satisfied by them. Then, some linear and quadratic relations are given between Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials.

scholarly journals On Generalized Jacobsthal and Jacobsthal-Lucas polynomials

2016 ◽  
Vol 24 (3) ◽  
pp. 61-78 ◽  
Author(s):  
Paula Catarino ◽  
Maria Luisa Morgado
Keyword(s):  
Generating Functions ◽  
Special Kind ◽  
Generalized Polynomials ◽  
Lucas Polynomials ◽  
Tridiagonal Matrices

AbstractIn this paper we introduce a generalized Jacobsthal and Jacobsthal-Lucas polynomials, Jh,nand jh,n, respectively, that consist on an extension of Jacobsthal's polynomials Jn(𝑥) and Jacobsthal-Lucas polynomials jn(𝑥). We provide their properties and a generalization of the usual identities. We also present, for each one of these generalized polynomials, their ordinary generating functions and matrices. In the last part of the paper, we present some special kind of tridiagonal matrices whose entries are elements of these generalized polynomials.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

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